Adaptive control with recursive identification for stochastic linear systems: Multivariable case

This paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to an unstable, inverse stable, multivariable linear stochastic system with unknown parameters, whenever that system satisfies a certain positive real condition on its (moving average) noise dynamics: (i) The adaptive control part of the algorithm stabilizes and asymptotically optimizes the behaviour of the system in the sense that the sample mean square variation of the output around a given demand level equals that of a minimum variance control strategy implemented with known parameters. This optimal behaviour is subject to an offset Tr [M], where M is the variance of a dither signal added to the control action in order to produce a "continually disturbed control". For M > 0, it is shown that the input-output process satisfies a persistent excitation property and hence, subject to a simple identifiability condition, the next property holds: (ii) The observed input and output of the controlled system are taken as inputs to an approximate maximum likelihood algorithm (AML) which generates strongly consistent estimates of the system's parameters.